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Goldstone mode singularities in O(n) models

Monte Carlo (MC) analysis of the Goldstone mode singularities for the transverse and the longitudinal correlation functions, behaving as G_{\perp}(k) \simeq ak^{-λ_{\perp}} and G_{\parallel}(k) \simeq bk^{-λ_{\parallel}} in the ordered phase at k -> 0, is performed in the three-dimensional O(n) models with n=2, 4, 10. Our aim is to test some challenging theoretical predictions, according to which the exponents λ_{\perp} and λ_{\parallel} are non-trivial (3/2<λ_{\perp}<2 and 0<λ_{\parallel}<1 in three dimensions) and the ratio bM^2/a^2 (where M is a spontaneous magnetization) is universal. The trivial standard-theoretical values are λ_{\perp}=2 and λ_{\parallel}=1. Our earlier MC analysis gives λ_{\perp}=1.955 \pm 0.020 and λ_{\parallel} about 0.9 for the O(4) model. A recent MC estimation of λ_{\parallel}, assuming corrections to scaling of the standard theory, yields λ_{\parallel} = 0.69 \pm 0.10 for the O(2) model. Currently, we have performed a similar MC estimation for the O(10) model, yielding λ_{\perp} = 1.9723(90). We have observed that the plot of the effective transverse exponent for the O(4) model is systematically shifted down with respect to the same plot for the O(10) model by Δλ_{\perp} = 0.0121(52). It is consistent with the idea that 2-λ_{\perp} decreases for large $n$ and tends to zero at n -> \infty. We have also verified and confirmed the expected universality of bM^2/a^2 for the O(4) model, where simulations at two different temperatures (couplings) have been performed.